A spanning bandwidth theorem in random graphs

Allen, Peter ORCID: 0000-0001-6555-3501, Böttcher, Julia ORCID: 0000-0002-4104-3635, Ehrenmüller, Julia, Schnitzer, Jakob and Taraz, Anusch (2022) A spanning bandwidth theorem in random graphs. Combinatorics Probability and Computing, 31 (4). 598 - 628. ISSN 0963-5483

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Abstract

The bandwidth theorem of Böttcher, Schacht and Taraz states that any n-vertex graph G with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). Recently, a subset of the authors proved a random graph analogue of this statement: for $p\gg \big(\tfrac{\log n}{n}\big)^{1/\Delta}$ a.a.s. each spanning subgraph G of G(n,p) with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all n-vertex k-colourable graphs H with maximum degree $\Delta$ , bandwidth o(n), and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper, we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in G contain many copies of $K_\Delta$ then we can drop the restriction on H that $Cp^{-2}$ vertices should not be in triangles.

Item Type:	Article
Official URL:	https://www.cambridge.org/core/journals/combinator...
Additional Information:	© 2021 The Authors
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited:	16 Mar 2022 12:12
Last Modified:	12 Dec 2024 02:52
URI:	http://eprints.lse.ac.uk/id/eprint/114370

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